Indecomposable modules for the dual immaculate basis of quasi-symmetric functions

We construct indecomposable modules for the 0-Hecke algebra whose characteristics are the dual immaculate basis of the quasi-symmetric functions

Immaculate basis of the non-commutative symmetric functions

We introduce a new basis of the non-commutative symmetric functions whose elements have Schur functions as their commutative images. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions and decompose Schur functions according to a signed combinatorial formula

A combinatorial interpretation of the noncommutative inverse Kostka matrix

We provide a combinatorial formula for the expansion of immaculate noncommutative symmetric functions into complete homogeneous noncommutative symmetric functions. To do this, we introduce generalizations of Ferrers diagrams which we call GBPR diagrams. We define tunnel hooks, which play a role similar to that of the special rim hooks appearing in the E\u{g}ecio\u{g}lu-Remmel formula for the symmetric inverse Kostka matrix. We extend this interpretation to skew shapes and fully generalize to define immaculate functions indexed by integer sequences skewed by integer sequences. Finally, as an application of our combinatorial formula, we extend Campbell's results on ribbon decompositions of immaculate functions to a larger class of shapes.Comment: 44 pages, 15 figures; corrected typos; revised arguments in Section 6, results unchange

Extended Schur functions and bases related By involutions

We introduce two new bases of QSym, the flipped extended Schur functions and the backward extended Schur functions, as well as their duals in NSym, the flipped shin functions and the backward shin functions. These bases are the images of the extended Schur basis and shin basis under the involutions $\rho$ and $\omega$ on the quasisymmetric and noncommutative symmetric functions, which generalize the classical involution $\omega$ on the symmetric functions. In addition, we prove a Jacobi-Trudi rule for certain shin functions using creation operators. We define skew extended Schur functions and skew-II extended Schur functions based on left and right actions of NSym and QSym respectively. We then use the involutions $\rho$ and $\omega$ to translate these and other known results to our flipped and backward bases.Comment: 28 page